\subsection{Variation method with close-channel restricted}

We start with three hyperfine species a, b, c, where a is the common species in two channel.  And the Hamiltonian is written in the form 
\begin{equation}\label{eq:091117:Hamiltonian}
\begin{split}
 H=&\sum_\vk\epsilon^a_\vk{}a^+_\vk{}a^{}_\vk+\sum_\vk\epsilon^b_\vk{}b^+_\vk{}b^{}_\vk+\sum_\vk\epsilon^c_\vk{}c^+_\vk{}c^{}_\vk\\
  &+\nth{2}\sum_{\vk\vk'}U_{\vk\vk'}a^+_\vk{}b^+_{-\vk}{}b^{}_{-\vk'}a^{}_{\vk'}
	+\nth{2}\sum_{\vk\vk'}V_{\vk\vk'}a^+_\vk{}c^+_{-\vk}{}c^{}_{-\vk'}a^{}_{\vk'}\\
 &+\nth{2}\sum_{\vk\vk'}Y_{\vk\vk'}a^+_\vk{}b^+_{-\vk}{}c^{}_{-\vk'}a^{}_{\vk'}
	+\nth{2}\sum_{\vk\vk'}Y^*_{\vk\vk'}a^+_{\vk'}{}c^+_{-\vk'}{}b^{}_{-\vk}a^{}_{\vk}
\end{split} 
\end{equation}
By the Hermition condition we have 
\begin{equation}
 U_{\vk'\vk}=U^*_{\vk\vk'},\qquad{} V_{\vk'\vk}=V^*_{\vk\vk'}
\end{equation}
Here (a,b) is the open channel while (a,c) is the close channel.  We start from the ansatz as 
\begin{equation}\label{eq:ansatz}
 \ket{\Psi}=\prod_\vk\br{u_\vk+v_\vk{}a^\dg_\vk{}b^\dg_{-\vk}+w\phi_\vk{}u_\vk{}a^\dg_\vk{}c^\dg_{-\vk}}\ket{0}
\end{equation}
Where $\ket{\phi}=\sum\phi_\vk{}a^\dg_\vk{}c^\dg_{-\vk}\ket{0}$ is the normalized close-channel bound state in resoance.  $\sum\abs{\phi_k}^2=1$ and $H^{close}\ket{\phi}=E\ket{\phi}$, $H^{close}=\sum_\vk\epsilon^a_\vk{}a^\dg_\vk{}a^{}_\vk+\sum_\vk\epsilon^c_\vk{}c^\dg_\vk{}c^{}_\vk+\nth{2}\sum_{\vk\vk'}V_{\vk\vk'}a^\dg_\vk{}c^\dg_{-\vk}{}c^{}_{-\vk'}a^{}_{\vk'}$.
Here we require $\abs{u_\vk}^2+\abs{v_\vk}^2+\abs{wu_\vk\phi_\vk}^2=1$ for simple normalization.  For all the interaction term, there are two types of contribution,
for example, 
\begin{equation*}
\av{U_{\vk\vk'}a^\dg_\vk{}b^\dg_{-\vk}{}b^{}_{-\vk'}a^{}_{\vk'}}
=\sum_{\vk}U_{\vk\vk}\abs{v_\vk}^2+\sum_{\vk\neq\vk'}U_{\vk\vk'}v^{}_{\vk'}u^*_{\vk'}u^{}_\vk{}v^*_\vk
\end{equation*}
The first term is the Hatree term and the second term is more interesting pairing term. 


And the free energy is 
\begin{equation}
 \begin{split}
  &F\equiv\av{H-\mu{}N}\\
    =&\sum(\xi^a_\vk+\xi^b_\vk)\abs{v_\vk}^2+\sum(\xi^a_\vk+\xi^c_\vk)\abs{w}^2\abs{u_\vk}^2\abs{\phi_\vk}^2\\
    &+\nth2\sum_{\vk}U_{\vk\vk}\abs{v_\vk}^2+\nth2\sum_{\vk\neq\vk'}U_{\vk\vk'}v^{}_{\vk'}u^*_{\vk'}u^{}_\vk{}v^*_\vk\\
    &+\nth2\sum_{\vk}V_{\vk\vk}\abs{u_\vk\phi_\vk{}w}^2
      +\nth2\sum_{\vk\neq\vk'}V_{\vk\vk'}\phi^{}_{\vk'}\phi^*_\vk\abs{u^{}_{\vk'}u^{}_\vk{}w}^2\\
    &+\nth2\sum_{\vk}Y_{\vk\vk}w\phi^{}_{\vk}{u^{}_{\vk}}v^*_\vk{}
      +\nth2\sum_{\vk\neq\vk'}Y_{\vk\vk'}w\phi^{}_{\vk'}\abs{u^{}_{\vk'}}^2v^*_\vk{}u^{}_\vk\\
    &+\nth2\sum_{\vk}Y^*_{\vk\vk}w^*\phi^{*}_{\vk}{u^{*}_{\vk}}v^{}_{\vk}{}
      +\nth2\sum_{\vk\neq\vk'}Y^*_{\vk\vk'}w^*\phi^{*}_{\vk}\abs{u^{}_{\vk}}^2v^{}_{\vk'}{}u^{*}_{\vk'}\\
 \end{split}
\end{equation}
Where 
\begin{equation*}
 \xi^a_\vk=\epsilon^a_\vk-\mu^a,\qquad\xi^b_\vk=\epsilon^b_\vk-\mu^b,\qquad\xi^c_\vk=\epsilon^c_\vk-\mu^b
\end{equation*}

 %I drops the Hatree term as this in some sense just shift the chemical potentials as it only relates to the density.  
